Influential Mathematicians: Maurice René Fréchet
Hello from our pre-course newsletter series aimed at highlighting mathematicians that have directly or indirectly contributed to the mathematical formalism of quantum mechanics. It is our humble attempt of taking any fear of mathematics out of your mind by showing that mathematicians are ordinary humans too!
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Going back to the topic of influential mathematicians, today I'll cover Maurice René Fréchet, a disciple of Jacques Hadamard - the mathematician for whom as some of you can guess the Hadamard quantum gate is named after!
Who was Maurice Fréchet?
The School of Mathematics and Statistics University of St Andrew has a pretty comprehensive bio on Maurice René Fréchet that I've summarised for you below.
Maurice Fréchet's parents were Jacques and Zoé Fréchet. It was a Protestant family and at the time Maurice was born his father was the director of a Protestant orphanage in Maligny. There were six children in the family, Maurice being the fourth. While he was still a very young child, his father Jacques Fréchet was appointed as head of a Protestant school in Paris and the family moved there with high expectations of a good future. However, political decisions were to severely damage the comfortable life of the Fréchet family.
Before Maurice was born the constitution of the Third Republic had been drawn up. Although the Republicans were divided into two wings, these were united on their attitude to the church's role in politics and education. One of the leaders Jules Ferry held major positions of power during 1880-85 and he brought in new laws to make primary education free, compulsory, and secular. Religious teaching in schools was replaced by "civic education" and from this time on French education was secularised. Jacques Fréchet lost his job as headmaster and was unemployed. Maurice's mother came to the financial rescue of the family by setting up a boarding house for foreigners. This had the benefit that Maurice grew up surrounded by those speaking foreign languages and he developed an international outlook which was to remain with him throughout his life. After the French education system settled down from the impact of the legal restraints put upon it, Jacques Fréchet was able to find a job teaching within the new secular system. It meant that the family never enjoyed the standards they might otherwise have expected, but nevertheless they did return to a stable financial situation.
Maurice entered secondary education at the Lycée Buffon in Paris. There he was taught mathematics by Hadamard who was a teacher at the school from 1890 to 1893 before being appointed professor at the University of Bordeaux in 1894. Hadamard immediately saw the mathematical potential of his young pupil and coached him on an individual basis. This continued after Hadamard moved to Bordeaux, for he wrote to Fréchet setting him mathematical problems, and corrected his work with severe criticisms if there were any errors. The relationship was one in which Fréchet was extremely grateful for the encouragement and guidance that he was receiving, but he admitted much later that he lived in continual fear of not being able to solve the problems he was set.
After leaving school, Fréchet undertook military service before, in 1900, entering the École Normale Supérieure in Paris. There he still worried over the decision on whether to specialise in physics or mathematics, and was eventually persuaded to specialise in mathematics, not because he did not enjoy physics just as much, but rather because further study of physics required him to take chemistry courses which he disliked. Even before he was awarded his Agrégation des Sciences Mathematiques in 1903, Fréchet began publishing short papers. By the end of 1903 he had four papers in print, three of which were four pages long. Seven further papers appeared in 1904, then remarkably eleven papers in 1905 as he undertook research for his doctorate under Hadamard's supervision. Contact with several American mathematicians who were in Paris, in particular Edwin Wilson, led to Fréchet publishing some of his early papers in American Mathematical Society publications (Edwin Wilson was editor of the Transactions of the American Mathematical Society from 1903). Another task undertaken by Fréchet around this time was writing up Borel's lectures for publication. Fréchet attended these lectures while an undergraduate and wrote up the lectures during the winter of 1903-04. The book ‘Leçons sur les fonctions de variables réelles et les développements en séries de polynômes’ was published in 1905.
Fréchet wrote an outstanding doctoral dissertation ‘Sur quelques points du calcul fonctionnel’ submitted on 2 April 1906. In it he introduced the concept of a metric space, although he did not invent the name 'metric space' which is due to Hausdorff. The thesis concerns 'functional operations' and 'functional calculus' and is developed from ideas due to Hadamard and Volterra. The importance of the thesis is that it develops axiomatic analysis systems providing an abstraction of different objects studied by analysis in a similar way to group theory providing an abstraction of algebraic systems. This parallel is drawn by Fréchet himself who requires sufficient structure on his abstract systems so that limits and continuity can be studied. He defines a functional operation as a numerically valued function defined on arbitrary objects which he wants to include points, lines, functions, numbers, surfaces etc. The functional calculus of his thesis is then the systematic study of functional operations.
A versatile mathematician, Fréchet served as professor of mathematics at the Lycée in Besançon (1907-08), professor of mathematics at the Lycée in Nantes (1908-09), then professor of mechanics at the Faculty of Science in Poitiers (1910-19). He married Suzanne Carrive in 1908 and they had four children; Hélène, Henri, Denise, and Alain. Fréchet had arranged to spend the academic year 1914-15 at the University of Illinois at Urbana in the United States and had accepted an appointment there for one year. He and his family were packed and ready to travel to the port to board their ship for the United States when World War I broke out and Fréchet was required for military service.
He was mobilised on 4 August 1914 but because of his language skills, initially gained when his mother ran the establishment of foreigners, he was attached to the British Army as an interpreter. This may have resulted in a slightly safer job than he would otherwise have had, but nevertheless he spent about two and a half years at or near the front, so was fortunate to survive. A great many French academics perished during the war, for the French belief in equality meant they tended to fight in the trenches rather than undertake specialised war work for which their expertise made them especially useful. In fact there is evidence that Fréchet had arranged with some American mathematicians to publish his complete works if he did not survive the war. That he chose to negotiate with Americans is almost certainly a sign that he felt more appreciated in that country than in his own. Not only did he undertake this dangerous work during the war, but Fréchet continued to produce frequent mathematics papers. One can only marvel at how he was able to continue with cutting edge research in such circumstances and with so little time to devote to his mathematics.
For the period of the war Fréchet retained his post at the Faculty of Science in Poitiers despite not being able to teach there. However before he was released from military service at the end of the war, he was selected to go to Strasbourg to assist with re-establishing the university there. He was both professor of higher analysis at the University of Strasbourg and Director of the Mathematics Institute there from 1919 to 1927. As he had been in earlier times, Fréchet was able to continue to produce a large research output despite heavy duties. He now had major administrative duties, one of the first being setting up and organising the International Congress of Mathematicians in Strasbourg in 1920. This was a difficult Congress for political reasons, since German and Austrian mathematicians were banned but this resulted in strong opinions and numerous arguments. An indication of his remarkable research output is that he had 36 papers published in the two years 1924 and 1925. It was after going to Strasbourg that he began to become interested in statistics but he only published a small number of articles on probability at this stage, most of his papers being on general analysis and topology. However, he taught courses on probability, statistics, and insurance mathematics at Strasbourg.
From November 1928 Fréchet held posts in Paris, but from this time on he concentrated more on statistics. It was Borel who encouraged Fréchet to seek positions in Paris and he supported his candidacy. There is also a suggestion that Fréchet had a difference of opinion with the Council of the Faculty of Science at Strasbourg which meant he was both pleased to return to Paris and not unhappy at leaving Strasbourg. He held several different positions in the field of mathematics in Paris between 1928 and 1948 when he retired. He was director of studies at the École des Hautes-Études, then professor at the Faculty of Science in Paris. From 1929 he was also professor of analysis and mechanics at the École Normale Supérieure.
As we have indicated, Fréchet made major contributions to the topology of point sets, and defined and founded the theory of abstract spaces. Fréchet also made important contributions to statistics, probability and calculus. There are different ways that people make major contributions to the progress of mathematics, some by solving the big questions, others by proposing new areas for research. Fréchet recognised himself that he fell into the latter category. In his dissertation of 1906, discussed above, he started a whole new area with his investigations of functionals on a metric space and formulated the abstract notion of compactness. In 1907 he discovered an integral representation theorem for functionals on the space of quadratic Lebesgue integrable functions. A similar result was discovered independently by Riesz. His introduction of general topology has been somewhat less appreciated than would otherwise have been the case since the publication of Hausdorff's major text in 1914 provided a more popular view.
Quantum Formalism Influence
I think it's fair to consider Fréchet as a godfather of Functional Analysis. The branch of mathematics that the general mathematical formalism of quantum mechanics is built upon. His foundational work on metric spaces and point-set topology opened the door for other mathematicians to create abstract concepts that are deployed in the quantum formalism. On a curious note, the reason that Dirac's bra-ket notation makes some mathematical sense is thanks to the Riesz–Fréchet representation theorem! :)
That’s it for today! I’ll end with a recommendation of some reading materials related to Fréchet:
Sur la loi de répartition de certaines grandeurs géographiques
"Les Probablititês Associées à un Système d'Événements Compatibles et Dépendants ; I. Événements en Nombre Fini Fixe by Maurice Fréchet
A side note recommendation
Quantum Revolution: Is this the Canadian Century? Our recent analysis of the Canadian quantum ecosystem: https://zaikucapital.substack.com/p/quantum-revolution-is-this-the-canadian
Author: Bambordé Baldé, co-founder at Zaiku Group. Please take our upcoming quantum lectures start date survey if you haven’t already done so - we’ll keep it up until the end of this month!
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